According to the principle of the transformation from the summary of traditional-cradle machine tool to the one of free-form machine tool-assuring the orientation between the cutter axis and the work-piece axis during the process, axis-motion formulas in the form of polynomial for free-form machine tool were fitted with the least square method. The procedure was accomplished via exemplifying No.116 gear generator for traditional-cradle machine tool, and deducing the coordinate transformation matrixes under these two kinds of machine tool. Correctness of the procedure is validated by an example, which shows that the coordinate transformation matrixes under these two kinds of machine tool coincide, the distance between the tooth surface virtually manufactured by axis-motion formulas in the software of CATIA and the theoretical points lies in the range from 0mm to 0.009mm, and the distance can be controlled to a smaller one. The influence of axis-motion error on tooth-surface error was investigated, and axis B was found to be the one influencing tooth surface most significantly. This partially provides theoretical basis for improving the manufacturing accuracy of free-form machine tool.